You could do what humans normally do when dealing with arbitrally long numbers - treat them as strings.
We don't normally use values higher than milion in daily life - would you dictate a 10 digit phone number* as
"Six bilions**, two hundred eighty three milions, one hundred eighty five thousands, three hundred seven"
"Six two eight ...
In the spirit of functional programming, you can add run length encoding.
Practically speaking, we see that there is no more effective of a method for naming numbers than giving them a sequence of digits. This is exactly what we do in mathematics. 513843843246213513513813502162324713513541263 is simply best described as that series of digits.
So the issue ...
A similar system to the one you describe is attested in Fijian. Fijian has a single-dual-paucal-plural distinction in its pronouns only. However, almost every sentence must contain a subject pronoun. I'm not sure about the status of imperatives in Fijian. Fijian also has VOS word order, where S is a lexical subject.
Independent nouns themselves are not ...
For precise integers, I see no viable systematic alternative that covers them all up to arbitrary size. But often, you don't need precise integers with all digits specified, spelled out, and spoken. For unprecise integers you can design some spoken form of exponent notation (like 1.0E9 in some programming language notation).
Both issues can be solved simultaneously.
Consider for a moment how you say 1 239 475 612 034 in English. It’s not (usually) said as:
one two three nine four seven five six one two zero three four
Instead, it’s read out as:
one trillion two hundred thirty nine billion four hundred seventy five million six hundred twelve thousand and thirty four
If -el "applies", I'm not sure about using it between digits. By that logic, KarelBis would be 4 applied to 2, so 2^4. Going that route, and assuming base 10 is the more common counting system in your world, you would need a word for 10, eg. Dix (from French). So 1001 would be DixelDixelDixUn: 10 * 10 * 10 + 1. Or these moderators could use prime ...
Also attested in English, just not to the extent of your example.
Dual number existed in nouns & pronouns and was lost in nouns by Primitive Germanic times. Its use continued into West Germanic & Old English first & second person pronouns:
s d pl
1 ic wit we
2 þu yit ye
An example from a different grammatical component, to widen the perspective.
In Czech, the former dual number has been retained as a special plural form for some paired body part nouns to distinguish them from their non-body-part meanings. For example, ucho means "ear" or "pot handle", uši - the former dual form - means "ears" (...
You can have some fun with this.
If the moderators resemble humans and are not good at repeating identical strings an exact number of times, then you can use something like Hebrew numerals or Greek numerals but with short syllables. The idea being that you effectively have multiple series of digits that cover different place value positions. Combining this ...
After thinking about this a bit, I realised that you are basically describing a minimal–augmented pronoun system. In a minimal–augmented system, each person comes in two pronominal forms: minimal, which is singlar for 1, 2 and 3, but dual for 1+2; and augmented, which is more than minimal. Slotting your pronominal table into this system gives the following: